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We prove that for any prime $p\geq 3$ the minimal exponential growth rate ofthe Baumslag-Solitar group $BS(1,p)$ and the lamplighter group$\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also showthat for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equalto the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group$\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter valuealso serves to show that the lower bound of A.Mann from [Mann, Journal ofAlgebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNNextensions is optimal.

Minimal exponential growth rates of metabelian Baumslag-Solitar groups and lamplighter groups